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A grand challenge facing molecular programmers is the rational development of fast, robust, and isothermal architectures akin to "chemical central processing units" that can sense (bio-)chemical signals from their environment, perform complex computation, and orchestrate a physical response in situ. DNA strand displacement systems (DSDs) remain a compelling candidate, but are hampered by spurious reaction pathways that lead to incorrect output. DSDs that utilize the systematic leakless motif can be made arbitrarily robust at the cost of increasing redundancy and network size (scaling), and thus a degradation of kinetic performance. Another class of architectures utilize DNA hybridization, extension, and signal production of entirely sequestered outputs via strand-displacing polymerases (SDPs) that have resulted in impressive demonstrations; however, they face similar challenges of aberrant behavior such as mis-priming by incorrect signals. Our work introduces a unified polymerase-dependent toehold-mediated strand displacement (PD-TMSD) architecture that integrates the programmed specificity of DSDs with the unique advantages of SDPs. This unification enables systems that can be made arbitrarily robust, at any concentration range, without increasing network size. We propose a number of gate designs and composition rules to compute arbitrary Boolean functions, emulate arbitrary chemical reaction networks, and explore time-bounded probabilistic computation made possible by certain classes of SDPs. Our theoretical exploration is backed by preliminary experimental demonstrations. This contribution was inspired by the belief that molecular programming can meet or exceed the complexity exhibited in biology if we embrace its best understood molecular machinery and couple it with systematic design principles built upon a strong theoretical foundation.